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Which sequences of holes are admissible for periodic homogenization with Neumann boundary condition?

Published online by Cambridge University Press:  15 August 2002

Alain Damlamian  and
Patrizia Donato
Alain Damlamian
Affiliation:
Laboratoire d'Analyse et de Mathématiques Appliquée, UMR 8050 du CNRS, Universités de Marne-la-Vallée et Paris 12 Val-de-Marne, Université Paris 12, 94010 Créteil Cedex France; damlamian@univ-paris12.fr.
Patrizia Donato
Affiliation:
Laboratoire de Mathématiques Raphaël Salem, Université de Rouen, Site Colbert, 76821 Mont-Saint-Aignan Cedex, France; donato@univ-rouen.fr. Université Paris VI, Laboratoire Jacques-Louis Lions, Boîte Courrier 187, 75252 Paris Cedex, France; donato@ann.jussieu.fr.
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Abstract

In this paper we give a general presentation ofthe homogenization of Neumann type problems in periodically perforateddomains, including the case where the shape of the reference hole varies with the sizeof the period (in the spirit of the construction of self-similar fractals).We shows that H 0-convergence holds under the extra assumption thatthere exists a bounded sequence of extension operators forthe reference holes. The general classof Jones-domains gives an example where this result applies. When this assumption fails, another approach, usingthe Poincaré–Wirtingerinequality is presented. A corresponding class where it appliesis that of John-domains, for which the Poincaré–Wirtinger constantis controlled.The relationship between these two kinds of assumptions is also clarified.

Keywords

Periodic homogenizationperforated domainsH 0-convergencePoincaré–Wirtinger inequalityJones domainsJohn domains.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 8: A tribute to JL Lions , 2002 , pp. 555 - 585
Copyright
© EDP Sciences, SMAI, 2002

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